Resonator, unit having resonator, oscillator having unit and electronic apparatus having oscillator

ABSTRACT

An electronic apparatus comprises a display portion and a plurality of oscillators, one of which is a quartz crystal oscillator comprising: a quartz crystal oscillating circuit comprising; an amplification circuit and a feedback circuit. For example, the feedback circuit is constructed by a quartz crystal resonator comprising vibrational arms and a base portion, and having novel shape and electrode construction. Also, the quartz crystal resonator is housed in a package. In addition, from a relationship of an amplification rate of the amplification circuit and a feedback rate of the feedback circuit, an output signal of the quartz crystal oscillating circuit having an oscillation frequency of a fundamental mode of vibration for the quartz crystal resonator can be provided with a frequency of high stability.

FIELD OF THE INVENTION

The present invention relates to a resonator, a unit having theresonator, an oscillator having the unit and an electronic apparatushaving the oscillator.

BACKGROUND OF THE INVENTION

There are many electronic apparatus comprising a display portion and aquartz crystal oscillator at least. For example, cellular phones,wristwatches, facsimiles, digital cameras and DVD recorders comprising aquartz crystal oscillator are well known. Recently, because of highstability for frequency, miniaturization and the light weight nature ofthese electronic apparatus, the need for an electronic apparatuscomprising a smaller quartz crystal oscillator with a frequency of highstability has arisen. For example, the quartz crystal oscillator havinga quartz crystal tuning fork resonator housed in a unit, which vibratesin a flexural mode, is widely used as a time standard in an electronicapparatus such as the cellular phones, the wristwatches, the facsimiles,the digital cameras and the DVD recorders.

Similar to this, the same need has also arisen for an electronicapparatus comprising a contour mode resonator such as alength-extensional mode quartz crystal resonator, a width-extensionalmode quartz crystal resonator and a Lame mode quartz crystal resonatoror a thickness shear mode quartz crystal resonator or a SAW (SurfaceAcoustic Wave) resonator or a resonator for sensing angular velocitymade of a piezoelectric material such as quartz crystal, lithiumtantalite (LiTaO₃), lithium niobate (LiNbO₃) and ceramics.

Heretofore, however, it has been impossible to obtain an electronicapparatus comprising a smaller quartz crystal oscillator with aminiature quartz crystal tuning fork resonator of the prior art, capableof vibrating in a flexural mode, and having a frequency of highstability, a small series resistance and a high quality factor. This isthe reason why, when miniaturized, the quartz crystal tuning forkresonator of the prior art, capable of vibrating in a flexural mode hasa smaller electromechanical transformation efficiency. As a result, theresonator has a frequency of low stability, a large series resistanceand a reduced quality factor.

Additionally, there has been a big problem in the quartz crystaloscillator of the prior art having the quartz crystal tuning forkresonator of the prior art, such that a frequency of a fundamental modeof vibration of the tuning fork resonator which is an output signal ofthe oscillator jumps to a second overtone mode of vibration thereof byshock or vibration.

Similarly, however, it has been impossible to obtain an electronicapparatus comprising a smaller quartz crystal oscillator with a contourmode resonator such as a length-extensional mode quartz crystalresonator, a width-extensional mode quartz crystal resonator and a Lamemode quartz crystal resonator or a thickness shear mode quartz crystalresonator or a SAW resonator or a resonator for sensing angular velocityhaving a frequency of high stability, a small series resistance and ahigh quality factor because, when miniaturized, each resonator has asmall electromechanical transformation efficiency, as a result, afrequency of low stability, a large series resistance and a low qualityfactor, and also is not strong against shock.

It is, therefore, a general object of the present invention to provideembodiments of a quartz crystal resonator, a quartz crystal unit, aquartz crystal oscillator and an electronic apparatus of the presentinvention, which overcome or at least mitigate one or more of the aboveproblems.

SUMMARY OF THE INVENTION

The present invention relates to a resonator, a unit, an oscillator andan electronic apparatus comprising a display portion and a plurality ofoscillators, one of which comprises a quartz crystal oscillatorcomprising a quartz crystal oscillating circuit having an amplificationcircuit and a feedback circuit, and in particular, relates to a quartzcrystal resonator capable of vibrating in a flexural mode, a quartzcrystal unit having the quartz crystal resonator and a quartz crystaloscillator having the quartz crystal unit and having an output signal ofa frequency of high stability for a fundamental mode of vibration of thequartz crystal resonator, and also to a quartz crystal oscillator havinga suppressed second overtone mode of vibration of the quartz crystalresonator, in addition, relates to a quartz crystal oscillatorcomprising an another contour mode resonator such as alength-extensional mode resonator, a width-extensional mode resonatorand a Lame mode resonator or a thickness shear mode resonator, each madeof quartz crystal or a SAW resonator or a piezoelectric resonator forsensing angular velocity. The quartz crystal oscillator is, therefore,available for the electronic apparatus requiring a miniature quartzcrystal oscillator with high time accuracy and shock proof.

It is an object of the present invention to provide a miniature quartzcrystal resonator, capable of vibrating in a flexural mode, and having ahigh electromechanical transformation efficiency.

It is an another object of the present invention to provide a miniaturequartz crystal unit with a quartz crystal resonator, capable ofvibrating in a fundamental mode of vibration of a flexural mode, andhaving a high electromechanical transformation efficiency.

It is a further object of the present invention to provide a quartzcrystal oscillator with a miniature quartz crystal resonator, capable ofvibrating in a flexural mode, and having a frequency of high stability,a small series resistance R₁ and a high quality factor Q₁, whose nominalfrequency for a fundamental mode of vibration is within a range of 10kHz to 200 kHz. Especially, a frequency of about 32.768 kHz is veryavailable for a time standard of a frequency signal.

It is a still another object of the present invention to provide anelectronic apparatus comprising a display portion and a plurality ofoscillators.

According to one aspect of the present invention, there is provided aquartz crystal resonator comprising: a plurality of vibrational arms,each of the vibrational arms having a first main surface and a secondmain surface and side surfaces; and a base portion to which thevibrational arms are attached, in which the resonator has apiezoelectric constant e′₁₂in the range of 0.1 C/m² to 0.19 C/m² in theabsolute value.

According to a second aspect of the present invention, there is provideda quartz crystal unit comprising: a quartz crystal resonator having abase portion and a plurality of vibrational arms attached to the baseportion; a case for housing the quartz crystal resonator; and a lid forcovering an open end of the case, each of the vibrational arms having afirst main surface and a second main surface opposite the first mainsurface and side surfaces, in which the quartz crystal resonator has acutting angle in the range of ZYlwt(−20° to +20°)/(−25° to +25°)/(−18°to +18°) and a piezoelectric constant e′₁₂ of the resonator is within arange of 0.1 C/m² to 0.19 C/m² in the absolute value.

According to a third aspect of the present invention, there is provideda quartz crystal oscillator comprising: a quartz crystal oscillatingcircuit comprising; an amplification circuit comprising a CMOS inverterand a feedback resistor, and a feedback circuit comprising a quartzcrystal resonator capable of vibrating in a flexural mode, a pluralityof capacitors and a drain resistor, the quartz crystal resonator beinghoused in a package comprising a case for housing the quartz crystalresonator and a lid for covering an open end of the case, andcomprising: a plurality of vibrational arms, each of the vibrationalarms having a first main surface and a second main surface opposite thefirst main surface and side surfaces; and a base portion to which thevibrational arms are attached, in which the quartz crystal resonator hasa cutting angle in the range of ZYlwt (−20° to +20°)/(−25° to+25°)/(−18° to +18°) and a piezoelectric constant e′₁₂ of the resonatoris within a range of 0.1 C/m² to 0.19 C/m² in the absolute value.

According to a fourth aspect of the present invention, there is providedan electronic apparatus comprising a display portion and a plurality ofoscillators, one of the oscillators being a quartz crystal oscillatorcomprising: a quartz crystal oscillating circuit comprising; anamplification circuit having a CMOS inverter and a feedback resistor,and a feedback circuit having a quartz crystal tuning fork resonatorcapable of vibrating in a flexural mode of an inverse phase, a pluralityof capacitors and a drain resistor, the quartz crystal tuning forkresonator comprising a tuning fork base and a plurality of tuning forkarms connected to the tuning fork base, each of the tuning fork armshaving a first main surface and a second main surface opposite the firstmain surface and side surfaces, the quartz crystal tuning fork resonatorbeing housed in a package comprising a case for housing the resonatorand a lid for covering an open end of the case, in which the quartzcrystal tuning fork resonator has a fundamental mode of vibration and asecond overtone mode of vibration and the amplification circuit of thequartz crystal oscillating circuit has negative resistances −RL₁ and−RL₂ for the fundamental mode of vibration and the second overtone modeof vibration of the quartz crystal tuning fork resonator, in which anabsolute value of the negative resistances is defined by |−RL₁| and|−RL₂| and a ratio of the |−RL₁| and R₁ is greater than that of the|−RL₂| and R₂, where R₁ and R₂ represent a series resistance of thefundamental mode of vibration and the second overtone mode of vibrationof the quartz crystal resonator, respectively, in which an output signalof the quartz crystal oscillating circuit has an oscillation frequencyof the fundamental mode of vibration of the quartz crystal tuning forkresonator and is a clock signal which is used to display time at adisplay portion of the electronic apparatus, and in which the quartzcrystal tuning fork resonator has a cutting angle in the range of ZYlwt(−20° to +20°)/(−25° to +25°)/(−18° to +18°) and a piezoelectricconstant e′₁₂ of the resonator is within a range of 0.1 C/m² to 0.19C/m² in the absolute value.

Preferably, the piezoelectric constant e′₁₂ is within a range of 0.12C/m² to 0.19 C/m² in the absolute value.

Preferably, mounting arms protruding from the base portion comprise two.

Preferably, each of the vibrational arms has a groove having a firststepped portion and a second stepped portion.

Preferably, each of the vibrational arms has a through groove.

Preferably, a merit value M₂ of a second overtone mode of vibration ofthe quartz crystal resonator having vibrational arms and a base portionis less than 30.

Preferably, the quartz crystal oscillator with the quartz crystalresonator is constructed so that a ratio of an amplification rate α₁ ofthe fundamental mode vibration and an amplification rate α₂ of thesecond overtone mode vibration of the amplification circuit is greaterthan that of a feedback rate β₂ of the second overtone mode vibrationand a feedback rate β₁ of the fundamental mode vibration of the feedbackcircuit, and a product of the amplification rate α₁ and the feedbackrate β₁ of the fundamental mode vibration is greater than 1.

The present invention will be more fully understood by referring to thefollowing detailed specification and claims taken in connection with theappended drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a general view of a quartz crystal plate from which a quartzcrystal resonator of the present invention is formed;

FIG. 2 shows a plan view of a quartz crystal resonator of a firstembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 3 shows an A–A′ cross-sectional view of the vibrational arms of thequartz crystal resonator in FIG. 2;

FIG. 4 shows an A–A′ cross-sectional view of another embodiment of thevibrational arms of the quartz crystal resonator in FIG. 2;

FIG. 5 shows a plan view of a quartz crystal resonator of a secondembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 6 shows a plan view of a quartz crystal resonator of a thirdembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 7 shows a B–B′ cross-sectional view of the vibrational arms of theresonator in FIG. 6;

FIG. 8 shows a plan view of a quartz crystal resonator of a fourthembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 9 shows a plan view of a quartz crystal resonator of a fifthembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 10 shows a plan view of a quartz crystal resonator of a sixthembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 11 shows a plan view of a quartz crystal resonator of a seventhembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 12 shows a plan view of a width-extensional mode quartz crystalresonator constructing an electronic apparatus of the present invention;

FIG. 13( a) and FIG. 13( b) show a plan view of a thickness shear modequartz crystal resonator constructing an electronic apparatus of thepresent invention and a F–F′ sectional view of the resonator;

FIG. 14 shows a plan view of a Lame mode quartz crystal resonatorconstructing an electronic apparatus of the present invention;

FIG. 15( a) and FIG. 15( b) show a plan view of a resonator for sensingangular velocity constructing an electronic apparatus of the presentinvention and a G–G′ sectional view of the resonator;

FIG. 16( a) and FIG. 16( b) show a plan view of a quartz crystalresonator of an eighth embodiment of the present invention andcomprising a quartz crystal tuning fork resonator, and a J–J′ sectionalview of the resonator;

FIG. 17 shows a plan view of a quartz crystal unit of a first embodimentof the present invention and omitting a lid;

FIG. 18 shows a plan view of a quartz crystal unit of a secondembodiment of the present invention and omitting a lid;

FIG. 19 shows a cross-sectional view of a quartz crystal unit of a thirdembodiment of the present invention;

FIG. 20 shows a cross-sectional view of a quartz crystal oscillator of afirst embodiment of the present invention;

FIG. 21 shows a diagram of an embodiment of a quartz crystal oscillatingcircuit constructing a quartz crystal oscillator of the presentinvention;

FIG. 22 shows a diagram of the feedback circuit of FIG. 21; and

FIG. 23 shows a block diagram of an embodiment of an electronicapparatus of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring now to the drawings, the embodiments of the present inventionwill be described in more detail.

FIG. 1 is a general view of a quartz crystal plate 1 from which a quartzcrystal resonator of the present invention is formed, and particularly,a relationship of cutting angles θ_(x), θ_(y) and θ_(z) of the quartzcrystal plate 1 and its coordinate system is illustrated in FIG. 1. Thecoordinate system has original point o, electrical axis x, mechanicalaxis y and optical axis z of quartz crystal and o-xyz is constructed.

First, a quartz crystal plate perpendicular to z axis, so called, Zplate quartz crystal is taken. The Z plate quartz crystal has adimension of Width W₀, length L₀ and thickness T₀ corresponding to arespective direction of x, y and z axes.

Next, this Z plate quartz crystal is, first, rotated with an angle θ_(y)about the y axis, second, rotated with an angle θ^(x) about x′ axiswhich is a new axis of the x axis, and third, rotated with an angleθ_(z) about z″ axis which is a new axis of the z axis, . In this case,each of the x, y and z axes changes to x″, y″ and z″ axes, respectively,because each axis is rotated twice about two axes. A quartz crystalresonator of the present invention is, therefore, formed from the quartzcrystal plate with the rotation angles.

In other words, according to an expression of IEEE notation, a cuttingangle of the quartz crystal resonator of the present invention can beexpressed by ZYlwt(θ_(y))/(θ_(x))/(θ_(z)), and each of the angles θ_(y),θ_(x), θ_(z) will be described later in detail according to resonatorsof the present invention.

FIG. 2 shows a plan view of a quartz crystal resonator 10 of a firstembodiment of the present invention and which is a quartz crystal tuningfork resonator. The resonator 10 comprises vibrational arms 20 and 31and a base portion 40 attached to the vibrational arms, and the baseportion 40 has mounting arms 36 and 37 protruding from the base portion,each of which is mounted on a mounting portion of a package comprising acase for housing the resonator and a lid for covering an open end of thecase. In addition, each of the vibrational arms 20 and 31 has a firstmain surface and a second main surface opposite the first main surfaceand side surfaces, and the vibrational arms 20 and 31 have grooves 21and 27, respectively, each of which has stepped portions comprising afirst stepped portion and a second stepped portion. Also, the resonator10 has cutting angles θ_(y), θ_(x) and θ_(z) which are within a range of−20° to +20°, −25° to +25° and −18° to +18°, respectively, namely, acutting angle of the resonator is within a range of ZYlwt(−20° to+20°)/(−25° to +25°)/(−18° to +18°). In this embodiment, the quartzcrystal tuning fork resonator can vibrate in a flexural mode of afundamental mode of an inverse phase, and which is one of a contour modequartz crystal resonator.

In more detail, the groove 21 is constructed to include a portion of acentral linear line 41 of the arm 20, and the groove 27 is similarlyconstructed to include a portion of a central linear line 42 of the arm31. Each of the grooves 21 and 27 has a width W₂, and the width W₂including a portion of the central linear lines 41 and 42, is preferablebecause a large moment of inertia occurs at the arms 20 and 31 and thearms can vibrate in a flexural mode easily. As a result, the quartzcrystal tuning fork resonator capable of vibrating in a fundamental modecan be obtained with a small series resistance R₁ and a high qualityfactor Q₁.

In addition, when each of the vibrational arms 20 and 31 has part widthsW₁ and W₃, an arm width W of the arms 20 and 31 has a relationship ofW=W₁+W₂+W₃, and the part widths W₁ and W₃ are constructed so that W₁≧W₃or W₁<W₃. In addition, the width W₂ is constructed so that W₂≧W₁, W₃. Inthis embodiment, also, the grooves are constructed at the arms so that aratio W₂/W of the width W₂ and the arm width W is greater than 0.35 andless than 1, preferably, within a range of 0.35 to 0.95 and a ratio t₁/tis less than 0.79, where t₁ and t are a thickness of the groove and thevibrational arms, as shown in FIG. 3, to obtain a very large moment ofinertia of the vibrational arms. That is, the quartz crystal tuning forkresonator, capable of vibrating in the fundamental mode, and having afrequency of high stability can be provided with a small seriesresistance R₁, a high quality factor Q₁ and a small capacitance ratio r₁because it has a very large electromechanical transformation efficiency.

Likewise, each of the vibrational arms 20 and 31 has a length L and eachof the grooves 21 and 27 has a length l₀ (not shown here). In thisembodiment, a ratio of the length l₀ and the length L is within a rangeof 0.3 to 0.8 to get a quartz crystal tuning fork resonator with seriesresistance R₁ of a fundamental mode of vibration smaller than seriesresistance R₂ of a second overtone mode of vibration. In other words,the length l₀ is within a range of 30% to 80% to the length L. Ingeneral, the length l₀ is within a range of 0.45 mm to 1.25 mm. Also,when a plurality of grooves are formed in at least one of upper andlower faces of the arms and divided into the length direction of thearms, the length l₀ is a total length of the grooves.

In addition, electrodes 25 and 26 are disposed on side surfaces of thevibrational arm 20 and an electrode 23 is disposed on a surface of thegroove 21, which extends into the mounting arm 36 having a connectingportion 34. Similar to this, electrodes 32 and 33 are disposed on sidesurfaces of the vibrational arm 31 and an electrode 29 is disposed on asurface of the groove 27, which extends into the mounting arm 37 havinga connecting portion 35. Each of the connecting portions 34 and 35 has alength L₂ and a width W_(S), and each of the mounting arms has a lengthL₃ and a width W₆. Also, the electrode 23 is connected to the electrodes32 and 33, and the electrode 29 is connected to the electrodes 25 and26.

In this embodiment, the length 10 of the groove corresponds to a lengthl_(d) of the electrode disposed inside each of the grooves, when thelength l_(d) of the electrode is less than the length l₀ of the groove,namely, the length l₀ is of the length l_(d) of the electrode. Inaddition, the base portion 40 has a length L₁ and a width W_(H), thelength L₁ is less than 0.5 mm, preferably, within a range of 0.015 mm to0.49 mm, and a total length L_(t) (=L+L₁) in this embodiment is lessthan 2.1 mm, preferably, within a range of 1.02 mm to 1.95 mm to obtaina miniature quartz crystal resonator. Also, when a distance W₄ betweenthe vibrational arms is taken, a total width W₅ (=2W+W₄) is less than0.53 mm, preferably, within a range of 0.15 mm to 0.52 mm, and the widthW₅ is equal to or less than the W_(H) which is less than 0.55 mm,preferably, within a range of 0.15 mm to 0.53 mm.

In addition, the length L₃ is greater than or equal to the length L₂ andalso, the length L₁ is greater than or equal to the length L₂ or thelength L₁ is less than the length L₂. In actual, a value of L₁−L₂ iswithin a range of −0.1 mm to 0.32 mm, especially, when W_(H) is greaterthan W₅, a distance L₄ between an edge of the connecting portion and anouter edge of the vibrational arm is within a range of 0.012 mm to 0.38mm. Also, the width W₆ is less than 0.45 mm and the length L₃ is lessthan 2.1 mm, preferably, within a range of 0.3 mm to 1.85 mm to reduce aleakage energy by vibration, and also, the width W_(S) is less than 0.41mm and the length L₂ is within a range of 0.04 mm to 0.5 mm to get ashock proof quartz crystal resonator having a reduced leakage energy byvibration.

Moreover, the distance W₄ and the width W₂ are constructed so thatW₄≧W₂, and more, the distance W₄ is within a range of 0.045 mm to 0.65mm and the width W₂ is within a range of 0.02 mm to 0.12 mm because itis easy to form a resonator shape and grooves formed at the vibrationalarms separately by a photo-lithographic process and an etching process,consequently, a frequency stability for a fundamental mode of vibrationof the resonator gets higher than that for a second overtone mode ofvibration thereof. In this embodiment, a quartz wafer having a thicknesst of 0.045 mm to 0.35 mm is used.

For example, in order to get a smaller-sized quartz crystal tuning forkresonator, capable of vibrating in a flexural mode, it is necessary thatthe width W₂ of the groove is less than 0.07 mm and the arm width W isless than 0.18 mm, and preferably, the W is greater than 0.05 mm andless than 0.1 mm. Also, the thickness t₁ of the groove is within a rangeof 0.01 mm to 0.085 mm approximately, and the part widths W₁ and W₃ areless than 0.021 mm, respectively, preferably, less than 0.015 mm. Inaddition, a groove provided on at least one of an obverse face and areverse face of the vibrational arms of this embodiment may be a throughhole, namely, the thickness of the hole t₁=0.

In more detail, to obtain a quartz crystal tuning fork resonator,capable of vibrating in a flexural mode, and having a frequency of highstability which achieves high time accuracy, it is necessary to obtainthe resonator whose resonance frequency is not influenced by shuntcapacitance because quartz crystal is a piezoelectric material and thestability for frequency is very dependent on the shunt capacitance. Inorder to decrease the influence on the resonance frequency by the shuntcapacitance, a merit value M_(i) plays an important role. Namely, themerit value M_(i) that expresses inductive characteristics, anelectromechanical transformation efficiency and a quality factor of aquartz crystal tuning fork resonator, is defined by a ratio Q_(i)/r_(i)of a quality factor Q_(i) and capacitance ratio r_(i), namely, M_(i) isgiven by M_(i)=Q_(i)/r_(i) where i shows a vibration order of theresonator, and for example, when i=1 and 2, merit values M₁ and M₂ arefor a fundamental mode of vibration of the resonator and a secondovertone mode of vibration thereof, respectively.

Also, a frequency difference Δf of resonance frequency f_(s) ofmechanical series independent on the shunt capacitance and resonancefrequency f_(r) dependent on the shunt capacitance is inverselyproportional to the merit value M_(i). The larger the value M_(i)becomes, the smaller the difference Δf becomes. Namely, the influence onthe resonance frequency f_(r) by the shunt capacitance decreases becauseit is close to the resonance frequency f_(s). Accordingly, the largerthe M_(i) becomes, the higher the stability for frequency of the quartzcrystal tuning fork resonator becomes because the resonance frequencyf_(r) of the resonator is almost never dependent on the shuntcapacitance. Namely, the quartz crystal tuning fork resonator can beprovided with a high time accuracy.

In detail, the quartz crystal tuning fork resonator can be obtained withthe merit value M₁ of the fundamental mode of vibration greater than themerit value M₂ of the second overtone mode of vibration by providing theabove-described tuning fork shape, grooves and dimensions. That is tosay, a relationship of M₁>M₂ is obtained. As an example, when aresonance frequency of a quartz crystal tuning fork resonator capable ofvibrating in a flexural mode is about 32.768 kHz for a fundamental modeof vibration and the resonator has a value of W₂/W=0.5, t₁/t=0.34 andl₁/1=0.48, though there is a distribution in production, the resonatorhas a merit value of M₁>65 for the fundamental mode of vibration and amerit value of M₂<30 for the second overtone mode of vibration,respectively.

Namely, the quartz crystal tuning fork resonator can be provided withhigh inductive characteristics, good electromechanical transformationefficiency (small capacitance ratio r₁ and small series resistance R₁)and a high quality factor. As a result, a stability for frequency of thefundamental mode of vibration becomes higher than that of the secondovertone mode of vibration, and simultaneously, the second overtone modeof vibration can be suppressed because capacitance ratio r₂ and seriesresistance R₂ of the second overtone mode of vibration become greaterthan capacitance ratio r₁ and series resistance R₁ of the fundamentalmode of vibration, respectively. In particular, r₂ has a value greaterthan 1500 in this embodiment.

Therefore, the resonator capable of vibrating in the fundamental modevibration can be provided with a high time accuracy because it has thefrequency of high stability. Consequently, a quartz crystal oscillatorcomprising the quartz crystal tuning fork resonator of this embodimentoutputs an oscillation frequency of the fundamental mode vibration as anoutput signal, and the frequency of the output signal has a very highstability, namely, excellent time accuracy. In other words, the quartzcrystal oscillator of this embodiment has a remarkable effect such thata frequency change by ageing becomes extremely small. Also, anoscillation frequency of the quartz crystal resonator of this embodimentis adjusted so that a frequency deviation is within a range of −100 PPMto +100 PPM to a nominal frequency, e.g. 32.768 kHz, after mounting iton a mounting portion of a case for housing the resonator.

In addition, the groove thickness t₁, shown in FIG. 3, of the presentinvention is the thinnest thickness of the grooves because quartzcrystal is an anisotropic material and the groove thickness t₁ has adistribution when it is formed by a chemical etching method. In detail,a groove shape of the sectional view of vibrational arms in FIG. 3 has arectangular shape, but the groove shape has an about U shape actually.In the above-described embodiments, though the grooves are constructedat the arms, this invention is not limited to this, namely, arelationship of the merit values M₁ and M₂ can be applied to theconventional quartz crystal tuning fork resonator and a relationship ofa quartz crystal oscillating circuit comprising an amplification circuitand a feedback circuit can be also applied to the conventional quartzcrystal tuning fork resonator to suppress a second overtone modevibration and to get a high frequency stability for a fundamental modeof vibration of the quartz crystal tuning fork resonator.

FIG. 3 shows an A–A′ cross-sectional view of the vibrational arms 20 and31 of the quartz crystal resonator 10 in FIG. 2, and electrodeconstruction within the grooves. The vibrational arm 20 has grooves 21and 22 cut into it, which include a portion of central linear line ofthe arm 20. The grooves 21 and 22 have a first set of electrodes 23 and24 of the same electrical polarity, while the side surfaces of the arm20 have a second set of electrodes 25 and 26 having an oppositeelectrical polarity to the first set of electrodes 23 and 24. Thevibrational arm 31 has grooves 27 and 28 constructed in a similar manneras the vibrational arm 20. The grooves 27 and 28 have a third set ofelectrodes 29 and 30 of the same electrical polarity, and the sidesurfaces of the vibrational arm 31 have a fourth set of electrodes 32and 33 with the opposite electrical polarity to the third electrodes 29and 30. The electrodes disposed on the vibrational arms 20 and 31 areconnected as shown in FIG. 3, namely, two electrode terminals ofdifferent electrical polarity C–C′ are obtained.

In detail, the first set of electrodes 23 and 24 disposed on the grooves21 and 22 of the vibrational arm 20 have the same electrical polarity asthe fourth set of electrodes 32 and 33 disposed on both side surfaces ofthe vibrational arm 31, while the second set of electrodes 25 and 26disposed on both side surfaces of the vibrational arm 20 have the sameelectrical polarity as the third set of electrodes 29 and 30 disposed onthe grooves 27 and 28 of the arm 31. When a direct voltage is appliedbetween the electrode terminals C–C′, an electric field Ex occurs alongthe arrow direction inside the vibrational arms 20 and 31. As theelectric field Ex occurs perpendicular to the electrodes disposed on thevibrational arms, as shown in the arrow signs, the electric field Ex hasa very large value and a large distortion occurs at the vibrationalarms. As a result, a quartz crystal tuning fork resonator capable ofvibrating in a flexural mode is obtained with a small series resistanceR₁ and a high quality factor Q because even when miniaturized there is avery large electromechanical transformation efficiency for theresonator.

FIG. 4 shows an A–A′ cross-sectional view of another embodiment of thevibrational arms 20 and 31 of the quartz crystal resonator 10 in FIG. 2.The vibrational arm 20 has a through groove 21 a, which include aportion of central linear line of the arm 20. The through groove 21 ahas a first set of electrodes 23 a and 24 a of the same electricalpolarity, while the side surfaces of the arm 20 have a second set ofelectrodes 25 and 26 having an opposite electrical polarity to the firstset of electrodes 23 a and 24 a. The vibrational arm 31 has a throughgroove 27 a constructed in a similar manner as the vibrational arm 20.The through groove 27 a has a third set of electrodes 29 a and 30 a ofthe same electrical polarity, and the side surfaces of the vibrationalarm 31 have a fourth set of electrodes 32 and 33 with the oppositeelectrical polarity to the third electrodes 29 a and 30 a. Theelectrodes disposed on the vibrational arms 20 and 31 are connected asshown in FIG. 4, namely, two electrode terminals of different electricalpolarity E–E′ are obtained.

In detail, the first set of electrodes 23 a and 24 a disposed on thethrough groove 21 a of the vibrational arm 20 have the same electricalpolarity as the fourth set of electrodes 32 and 33 disposed on both sidesurfaces of the vibrational arm 31, while the second set of electrodes25 and 26 disposed on both side surfaces of the vibrational arm 20 havethe same electrical polarity as the third set of electrodes 29 a and 30a disposed on the through groove 27 a of the arm 31. When a directvoltage is applied between the electrode terminals E–E′, an electricfield Ex occurs along the arrow direction inside the vibrational arms 20and 31. As the electric field Ex occurs perpendicular to the electrodesdisposed on the vibrational arms, as shown in the arrow signs, theelectric field Ex has a very large value and a large distortion occursat the vibrational arms. As a result, a quartz crystal tuning forkresonator capable of vibrating in a flexural mode is obtained with asmall series resistance R₁ and a high quality factor Q because even whenminiaturized there is a very large electromechanical transformationefficiency for the resonator.

FIG. 5 shows a plan view of a quartz crystal resonator 50 of a secondembodiment of the present invention, and which is a quartz crystaltuning fork resonator capable of vibrating in a flexural mode. Theresonator 50 comprises vibrational arms 60 and 71 and a base portion 80attached to the vibrational arms, and the base portion 80. has amounting arm 77 protruding from the base portion, and the mounting arm77 is mounted on a mounting portion of a package comprising a case forhousing the resonator and a lid for covering an open end of the case. Inaddition, each of the vibrational arms 60 and 71 has a first mainsurface and a second main surface and side surfaces, and the vibrationalarms 60 and 71 have grooves 61 and 67, respectively, each of which hasstepped portions comprising a first stepped portion and a second steppedportion. Also, the resonator 50 has the same cutting angles θ_(y), θ_(x)and θ_(z) and the same dimensions W₁, W₂, W₃, W₄, W₅, W, L₁, L₂, L₃, L₄and L as the resonator of FIG. 2.

In more detail, the groove 61 is constructed to include a portion of acentral linear line 81 of the arm 60, and the groove 67 is similarlyconstructed to include a portion of a central linear line 82 of the arm71. Each of the grooves 61 and 67 has a width W₂, and the width W₂includes a portion of the central linear lines 81 and 82 because a largemoment of inertia occurs at the arms 60 and 71. In this embodiment, theresonator is a quartz crystal tuning fork resonator capable of vibratingin a flexural mode and which can vibrate in a fundamental mode of aninverse phase easily. As a result, the quartz crystal tuning forkresonator capable of vibrating in a fundamental mode of an inverse phasecan be obtained with a small series resistance R₁ and a high qualityfactor Q₁.

In addition, electrodes 65 and 66 are disposed on side surfaces of thevibrational arm 60 and an electrode 63 is disposed on a surface of thegroove 61, and which is connected to electrodes 72 and 73 disposed onside surfaces of the vibrational arm 71. Similar to this, an electrode69 is disposed on a surface of the groove 67, which extends into themounting arm 77 having a connecting portion 75. Also, the electrode 69is connected to the electrodes 65 and 66.

FIG.6 shows a plan view of a quartz crystal resonator 90 of a thirdembodiment of the present invention, which is a quartz crystal tuningfork resonator capable of vibrating in a flexural mode. The resonator 90comprises vibrational arms 91 and 92 and a base portion 95 attached tothe vibrational arms, and the base portion 90 has a mounting arm 96protruding from the base portion. In this embodiment, the mounting arm96 is between the vibrational arms 91 and 92 and is mounted on amounting portion of a package comprising a case for housing theresonator and a lid for covering an open end of the case. In addition,each of the vibrational arms 91 and 92 has a first main surface and asecond main surface opposite the first main surface and side surfaces,and the vibrational arms 91 and 92 have grooves 93 and 94, respectively,each of which has stepped portions comprising a first stepped portionand a second stepped portion. Also, the resonator 90 has the samecutting angles θ_(y), θ_(x) and θ_(z) as the quartz crystal resonator 10of FIG. 2. In this embodiment, the quartz crystal tuning fork resonatorcan vibrate in a flexural mode of a fundamental mode of an inversephase.

In detail, similar to the resonator of FIG. 2, the groove 93 is alsoconstructed to include a portion of a central linear line of the arm 91,and also, the groove 94 is similarly constructed to include a portion ofa central linear line of the arm 92. Each of the grooves 93 and 94 has awidth W₂, and the width W₂ includes a portion of the central linearlines because a large moment of inertia occurs at the vibrational arms91 and 92 and which can vibrate in a flexural mode easily. As a result,the quartz crystal tuning fork resonator capable of vibrating in afundamental mode can be obtained with a small series resistance R₁ and ahigh quality factor Q₁.

In addition, the base portion 95 has a length L₁ and a width W_(H).Also, each of the vibrational arms 91 and 92 has a width W, part widthsW₁and W₃ and a width W₂ of the groove, namely, there is a relationshipof W=W₁+W₂+W₃. In detail, the resonator 90 has the same dimensions L₁,W_(H), W₁, W₂, W₃, and W and the same relationship as the resonator 10of FIG. 2. Moreover, the mounting arm 96 has a width W₁₁, and there is adistance W₁₀ between the vibrational arm 91 or the vibrational arm 92and the mounting arm 96. In order to get a quartz crystal tuning forkresonator with reduced leakage energy by vibration of the vibrationalarms, the distance W₁₀ is within a range of 0.032 mm to 0.21 mm and thewidth W₁₁ is within a range of 0.21 mm to 0.88 mm. In addition, to get ashockproof quartz crystal tuning fork resonator, W₁₁ has a relationshipof (1.2 to 7.6)×W. In this embodiment, a total width W_(t)(=2W+2W₁₀+W₁₁)is less than 1.3 mm, preferably, within a range of 0.52 mm to 1.2 mm, toget a miniature quartz crystal tuning fork resonator.

FIG. 7 shows a B–B′ cross-sectional view of the vibrational arms 91 and92 of the resonator 90 in FIG. 6. The vibrational arm 91 has grooves 93and 97 cut into it, and the grooves 93 and 97 have a first set ofelectrodes 100 and 101 of the same electrical polarity, while the sidesurfaces of the arm 91 have a second set of electrodes 99 and 102 havingan opposite electrical polarity to the first set of electrodes 100 and101. The vibrational arm 92 has grooves 94 and 98 constructed in asimilar manner as the vibrational arm 91. The grooves 94 and 98 have athird set of electrodes 106 and 107 of the same electrical polarity, andthe side surfaces of the vibrational arm 92 have a fourth set ofelectrodes 105 and 108 with the opposite electrical polarity to thethird electrodes 106 and 107. The electrodes disposed on the vibrationalarms 91 and 92 are connected as shown in FIG. 7, namely, two electrodeterminals of different electrical polarity D–D′ are obtained.

In detail, the first set of electrodes 100 and 101 disposed on thegrooves 93 and 97 of the vibrational arm 91 have the same electricalpolarity as the fourth set of electrodes 105 and 108 disposed on bothside surfaces of the vibrational arm 92 and an electrode 104 disposed onthe mounting arm 96, while the second set of electrodes 99 and 102disposed on both side surfaces of the vibrational arm 91 have the sameelectrical polarity as the third set of electrodes 106 and 107 disposedon the grooves 94 and 98 of the arm 92 and an electrode 103 disposed onthe mounting arm 96. When a direct voltage is applied between theelectrode terminals D–D′, an electric field Ex occurs along the arrowdirection inside the vibrational arms 91 and 92. As the electric fieldEx occurs perpendicular to the electrodes disposed on the vibrationalarms, as shown in the arrow signs, the electric field Ex has a verylarge value and a large distortion occurs at the vibrational arms.Consequently, a quartz crystal tuning fork resonator capable ofvibrating in a flexural mode is obtained with a small series resistanceR₁ and a high quality factor Q because even when miniaturized, there isvery large electromechanical transformation efficiency for theresonator.

FIG. 8 shows a plan view of a quartz crystal resonator 110 of a fourthembodiment of the present invention, which is a quartz crystal tuningfork resonator. The resonator 110 comprises vibrational arms 111 and 112and a base portion 113 attached to the vibrational arms, and the baseportion 113 has mounting arms 114 and 115 protruding from the baseportion, each of which is mounted on a mounting portion of a packagecomprising a case for housing the resonator and a lid for covering anopen end of the case. In this embodiment, the vibrational arms 111 and112 have grooves 116 and 117, respectively. In more detail, each of thevibrational arms 111 and 112 has an end portion connected to the baseportion and a free end portion, when a distance measured from the endportion to the free end portion is a length L, each of the vibrationalarms has a width W between the end portion and half a length L/2 of eachof the vibrational arms and a width W_(e) between half the length L/2and the length L of the free end portion of each of the vibrationalarms, and the width W is less than the width W_(e). In this embodiment,the quartz crystal tuning fork resonator can vibrate in a flexural modeand vibrate in a fundamental mode of an inverse phase.

Similar to this, a quartz crystal tuning fork resonator 120 of a fifthembodiment of the present invention is shown in FIG. 9 showing a planview thereof. The resonator 120 comprises vibrational arms 121 and 122and a base portion 123 attached to the vibrational arms, and the baseportion 123 has a mounting arm 124 protruding from the base portion. Inthis embodiment, the mounting arm 124 is between the vibrational arms121 and 122 and the vibrational arms 121 and 122 have grooves 125 and126, respectively. For this case the width W is also less than the widthWe similar to that of FIG. 8.

FIG. 10 shows a plan view of a quartz crystal resonator 130 of a sixthembodiment of the present invention, which is a quartz crystal tuningfork resonator. The resonator 130 comprises vibrational arms 131 and 132and a base portion 133 attached to the vibrational arms, and the baseportion 133 has mounting arms 134 and 135 protruding from the baseportion, each of which is mounted on a mounting portion of a packagecomprising a case for housing the resonator and a lid for covering anopen end of the case. In this embodiment, the vibrational arms 131 and132 have grooves 136 and 137, respectively. In more detail, each of thevibrational arms 131 and 132 has an end portion connected to the baseportion and a free end portion, when a distance measured from the endportion to the free end portion is a length L, each of the vibrationalarms has a width W between the end portion and half a length L/2 of eachof the vibrational arms and a width W_(e) between half the length L/2and the length L of the free end portion of each of the vibrationalarms, and the width W is less than the width W_(e). In this embodiment,the quartz crystal tuning fork resonator can vibrate in a flexural modeand vibrate in a fundamental mode of an inverse phase.

Similar to this, a quartz crystal tuning fork resonator 140 of a seventhembodiment of the present invention is shown in FIG. 11 showing a planview thereof. The resonator 140 comprises vibrational arms 141 and 142and a base portion 143 attached to the vibrational arms, and the baseportion 143 has a mounting arm 144 protruding from the base portion. Inthis embodiment, the mounting arm 144 is between the vibrational arms141 and 142 and the vibrational arms 141 and 142 have grooves 145 and146, respectively. For this case the width W is also less than the widthWe similar to that of FIG. 10.

Especially, the quartz crystal tuning fork resonator of the fourthembodiment to the seventh embodiment can be miniaturized with a smallseries resistance R₁ and a high quality factor Q₁ because it has a widthW_(e) greater than a width W, and the width W_(e) operates as a mass.

Next, a value of a piezoelectric constant e′₁₂ is described, which is ofgreat importance and necessary to excite a quartz crystal tuning forkresonator capable of vibrating in a flexural mode of the presentinvention. The larger a value of the piezoelectric constant e′₁₂becomes, the higher electromechanical transformation efficiency becomes.The piezoelectric constant e′₁₂ of the present invention can be definedby a function of the cutting angles θ_(y), θ_(x) and θ_(z) shown in FIG.1, and piezoelectric constants e₁₁=0.171 C/m² and e₁₄=−0.0406 C/m² ofquartz crystal. In order to obtain a quartz crystal tuning forkresonator, capable of vibrating in a flexural mode and having a smallseries resistance R₁ and a high quality factor Q, the piezoelectricconstant e′₁₂ of the present invention is within a range of 0.1 C/m² to0.19 C/m² in the absolute value. It is, therefore, easily understoodthat this value is enough large to obtain the quartz crystal tuning forkresonator with a small series resistance R₁ and a high quality factor Q.

Especially, in order to obtain a quartz crystal tuning fork resonatorcapable of vibrating in a flexural mode with a much smaller seriesresistance R₁, the piezoelectric constant e′₁₂ is preferably within arange of 0.12 C/m² to 0.19 C/m² in the absolute value.

In addition, as an example, a quartz crystal tuning fork resonatorcomprises a plurality of vibrational arms having a first vibrational armand a second vibrational arm, and a groove having a first steppedportion and a second stepped portion is formed in at least one of afirst main surface and a second main surface of each of the first andsecond vibrational arms, in which a first electrode is disposed on thefirst stepped portion of the groove, a second electrode is disposed onthe second stepped portion of the groove, and a third electrode isdisposed on each of side surfaces of each of the first and secondvibrational arms, in which the piezoelectric constant e′₁₂(=e′_(12i)) isbetween the first electrode and the third electrode disposed opposite tothe first electrode, and the piezoelectric constant e′₁₂(=e′_(12o)) isbetween the second electrode and the third electrode disposed oppositeto the second electrode, and in which the piezoelectric constantse′_(12i) and e′_(12o) are within the range of 0.12 C/m² to 0.19 C/m² inthe absolute value, respectively, and a product of the e′_(12i) and thee′_(12o) is greater than 0.

As an another example, a quartz crystal tuning fork resonator comprisesa plurality of vibrational arms having a first vibrational arm and asecond vibrational arm, and a through hole having a first side surfaceand a second side surface is formed in each of the first and secondvibrational arms, in which a first electrode is disposed on the firstside surface of the through hole, a second electrode is disposed on thesecond side surface of the through hole, and a third electrode isdisposed on each of side surfaces of each of the first and secondvibrational arms, in which the piezoelectric constant e′₁₂(=e′_(12i)) isbetween the first electrode and the third electrode disposed opposite tothe first electrode, and the piezoelectric constant e′₁₂(=e′_(12o)) isbetween the second electrode and the third electrode disposed oppositeto the second electrode, and in which the piezoelectric constantse_(12i) and e_(12o) are within the range of 0.12 C/m² to 0.19 C/m² inthe absolute value, respectively, and a product of the e′_(12i) and thee′_(12o) is greater than 0.

Therefore, the quartz crystal tuning fork resonator described above hasa small series resistance R₁ and a high quality factor Q, and also afrequency of high stability.

FIG. 12 shows a plan view of a width-extensional mode quartz crystalresonator 150 constructing an electronic apparatus of the presentinvention, and which vibrates in a width-extensional mode. Thewidth-extensional mode resonator 150 comprises a vibrational portion151, connecting portions 152, 152 a and supporting portions 153 and 154connected to a mounting portion 155 constructing the supportingportions. Namely, the vibrational portion 151 is connected to thesupporting portions 153, 154 having the mounting portion 155 through theconnecting portions 152, 152 a. In addition, an electrode 151 a isdisposed on an obverse surface of the vibrational portion 151 and anelectrode 151 b (not visible) is disposed on a reverse surface of thevibrational portion 151.

In more detail, the electrode 151 a disposed on the obverse surface ofthe vibrational portion 151 is connected to an electrode 153 a disposedon the mounting portion 155, while the electrode 151 b disposed on thereverse surface of the vibrational portion 151 is connected to anelectrode 154 a disposed on the mounting portion 155 through anelectrode 154 b disposed on a side surface of the mounting portion.Namely, a pair of electrodes is disposed on the obverse and reversesurfaces of the vibrational portion 151. Also, the vibrational portion151 has a width W₀ and a length L₀. In general, a ratio W₀/L₀ is lessthan 0.35. In addition, the mounting portion 155 is mounted on amounting portion of a package comprising a case for housing theresonator and a lid for covering an open end of the case. In thisembodiment, a cutting angle of the resonator is within a range of ZYlwt(80° to 100°)/(−10° to +10°)/(75° to +115°) and a piezoelectric constante′₃₁ of the resonator is within a range of 0.1 C/m² to 0.19 C/m² in theabsolute value to obtain the resonator with a small series resistance R₁and a high quality factor Q.

Similar to this, a length-extensional mode quartz crystal resonator canbe obtained by replacing the width W₀ with the length L₀. In this casethe connecting portions are formed in the width direction. In thisembodiment, a cutting angle of the length-extensional mode resonator iswithin a range of ZYlwt (80° to 100°)/(−10° to +10°)/(−35° to +35°) anda piezoelectric constant e′₃₂ of the resonator is within a range of 0.12C/m² to 0.19 C/m² in the absolute value to obtain the resonator with asmall series resistance R₁ and a high quality factor Q.

FIG. 13( a) and FIG. 13( b) show a plan view of a thickness shear modequartz crystal resonator 156 constructing an electronic apparatus of thepresent invention and a F–F′ sectional view of the thickness shear moderesonator 156 capable of vibrating in a thickness shear mode. Theresonator 156 comprises a vibrational portion 157 having a width W₀, alength L₀ and a thickness T₀, and electrodes 158 and 159 are disposed onan obverse surface and a reverse surface so that the electrodes haveopposite electrical polarity each other. Namely, a pair of electrodes isdisposed on the vibrational portion 157. Also, the resonator 156 ishoused in a package comprising a case for housing the resonator and alid for covering an open end of the case. In this embodiment, a cuttingangle of the resonator is within a range of ZYlwt (−5° to +5°)/±(37° to58°)/(85° to 95°) and a piezoelectric constant e′₃₄ of the resonator iswithin a range of 0.055 C/m² to 0.14 C/m² in the absolute value toobtain the resonator with a small series resistance R₁ and a highquality factor Q. In order to obtain much smaller series resistance R₁,the piezoelectric constant e′₃₄ of the resonator is preferably within arange of 0.085 C/m² to 0.12 C/m² in the absolute value.

FIG. 14 shows a plan view of a Lame mode quartz crystal resonator 210constructing an electronic apparatus of the present invention, andvibrating in a Lame mode. The Lame mode resonator 210 comprises avibrational portion 211, connecting portions 212, 213 and supportingportions 214, 215 having mounting portions 216 and 217. Namely, thevibrational portion 211 is connected to the supporting portions 214, 215through the connecting portions 212, 213. In addition, an electrode 218is disposed on an obverse surface of the vibrational portion 211 and anelectrode 219 (not visible) is disposed on a reverse surface of thevibrational portion 211.

In more detail, the electrode 218 disposed on the obverse surface of thevibrational portion 211 is extended into the mounting portion 217, whilethe electrode 219 disposed on the reverse face of the vibrationalportion 211 is extended into the mounting portion 216. Namely, a pair ofelectrodes is disposed on the obverse and reverse surfaces of thevibrational portion 151. Also, the vibrational portion 211 has a widthW₀ and a length L₀. In general, a ratio L₀/W₀ is approximately equal tom for a fundamental mode of vibration and an overtone mode of vibrationof the resonator, where m is an order of vibration of the resonator andan integer. For example, when a Lame mode quartz crystal resonator hasone of m=1, 2, 3 and n, the resonator vibrates in a fundamental mode form=1, a second overtone mode for m=2, a third overtone mode for m=3 andan n^(th) overtone mode for m=n.

Also, the m has a close relationship to the number of electrodesdisposed on the vibrational portion. For example, when an electrode isdisposed opposite each other on both of an obverse surface and a reversesurface of the vibrational portion, this is called “the number of twoelectrodes”, in other words, “a pair of electrodes”. Namely, thevibrational portion has the number of p electrodes, where p is an evennumber such as 2, 4, 6, 8 and 10. As an example, when p of thevibrational portion comprises 6, the vibrational portion vibrates in athird overtone mode. In this example, three pairs of electrodes aredisposed on the vibrational portion. Namely, the third overtone mode ofvibration is a principal vibration. As is apparent from thisrelationship, there is a relationship of m=p/2.

Therefore, the principal vibration vibrates in the order of vibrationcorresponding to the number of an electrode pair or electrode pairs. Forexample, the principal vibration vibrates in a fundamental mode ofvibration, a second overtone mode of vibration and a third overtone modeof vibration, respectively, for an electrode pair, two electrode pairsand three electrode pair disposed on the vibrational portion. In detail,when m_(e) electrode pairs are disposed on the vibrational portion, theprincipal vibration vibrates in an n^(th) overtone mode of vibration,and m_(e) corresponds to the n, where me is an integer. It is needlessto say that this concept can be applied to a width-extensional modequartz crystal resonator and a length-extensional mode quartz crystalresonator.

In more detail, an even number of electrodes are disposed on the obverseand reverse surfaces of the vibrational portion and the electrodesdisposed opposite each other has an opposite electrical polarity. Inaddition, each of the mounting portions 216, 217 is mounted on amounting portion of a package comprising a case for housing theresonator and a lid for covering an open end of the case. In thisembodiment, a cutting angle of the resonator is within a range of ZYlwt(−5° to +5°)/±(35° to 60°)/±(40° to 50°) and a piezoelectric constante′₃₂ of the resonator is within a range of 0.045 C/m² to 0.13 C/m² inthe absolute value to obtain the resonator with a small seriesresistance R₁ and a high quality factor Q.

FIG. 15( a) and FIG. 15( b) show a plan view of a resonator for sensingangular velocity constructing an electronic apparatus of the presentinvention and a G–G′ sectional view of the resonator. In thisembodiment, the resonator 220 comprises a quartz crystal tuning forkresonator, capable of vibrating in a flexural mode and comprisingvibrational arms 221, 222 and a base portion 223 attached to thevibrational arms, the base portion 223 is mounted on a mounting portionof a package comprising a case for housing the resonator and a lid forcovering an open end of the case. In addition, each of the vibrationalarms 221 and 222 has a first main surface and a second main surfaceopposite the first main surface and side surfaces, and the vibrationalarm 221 has grooves 224, 230, while the vibrational arm 222 has grooves225, 237, each of which has stepped portions comprising a first steppedportion and a second stepped portion. Also, a cutting angle of theresonator is within a range of ZYlwt (−20° to +20°)/(−25° to +25°)/(−18°to +18°) and a piezoelectric constant e′₁₂ of the resonator is within arange of 0.1 C/m² to 0.19 C/m² in the absolute value. The resonator ofthis embodiment can vibrate in a flexural mode of a fundamental mode andan inverse phase.

In more detail, electrodes 226 and 227 which are of the same electricalpolarity are disposed on the side surfaces such that an electrodeterminal H is defined, while an electrode 228 is disposed inside thegroove 224 and an electrode 229 which is of the same electrical polarityto the electrode 228 is disposed inside the groove 230 such that anelectrode terminal H′ of opposite electrical polarity to the electrodeterminal H is defined. Namely, two electrode terminals H–H′ for an inputsignal are constructed. On the other hand, electrodes 231, 232, 233which are of the same electrical polarity are disposed on the sidesurfaces and inside the grooves 225, 237 such that an electrode terminalI is defined, while electrodes 234, 235, 236 which are of the sameelectrical polarity are disposed on the side surfaces and inside thegrooves 225, 237 such that an electrode terminal I′ of oppositeelectrical polarity to the electrode terminal I is defined. Namely, twoelectrode terminals I–I′ for an output signal are constructed. Theresonator of this embodiment is made of quartz crystal, but may be madeof a piezoelectric material such as lithium tantalite, lithium niobateand ceramics.

FIG. 16( a) and FIG. 16( b) show a plan view and a J–J′ sectional viewof a quartz crystal resonator 230 of an eighth embodiment of the presentinvention, and which is a quartz crystal tuning fork resonator. Theresonator 230 comprises vibrational arms 231 and 232 and a base portion233 attached to the vibrational arms. In addition, each of thevibrational arms 231 and 232 has a first main surface and a second mainsurface opposite the first main surface and side surfaces, and the firstand second main surfaces have central linear portions 242, 243,respectively. The vibrational arm 231 has grooves 234, 235 and thevibrational arm 232 has grooves 235, 237. In addition, the grooves 234and 236 are formed between the central linear portion 242 and an outeredge of the vibrational arm 231, respectively. Similar to this, thegrooves 235 and 237 are formed between the central linear portion 243and an outer edge of the vibrational arm 232, respectively.

Moreover, each of the grooves 234, 235, 236 and 237 has a width W₈, anda width W₇ including a portion of the central linear lines 242 and 243is formed in each of the vibrational arms 231 and 232. In addition, adistance W₉ is formed in the width direction of the vibrational arms 231and 232 between an outer edge of the groove and an outer edge of thevibrational arms. In detail, a width W of the arms 231 and 232 hasgenerally a relationship of W=W₇+2W₈+2W₉, and the width W₈ isconstructed so that W₈≧W₇, W₉. In this embodiment, also, the grooves areconstructed at the arms so that a ratio W₈/(W/2) of the width W₈ and ahalf width of the width W is greater than 0.35 and less than 1,preferably, within a range of 0.35 to 0.95. In addition, the width W₇ isless than 0.05 mm, preferably, less than 0.03 mm and the width W₈ iswithin a range of 0.015 mm to 0.05 mm to obtain a very large moment ofinertia of the vibrational arms. That is, the quartz crystal tuning forkresonator, capable of vibrating in a fundamental mode, and having afrequency of high stability can be provided with a small seriesresistance R₁, a high quality factor Q₁ and a small capacitance ratio r₁because it has a very large electromechanical transformation efficiency.

In FIG. 16( b), the vibrational arm 231 has grooves 234, 236, 238 and240 cut into it. The grooves 234, 236, 238 and 240 have a first set ofelectrodes 256, 257, 258 and 259 of the same electrical polarity, whilethe side surfaces of the arm 231 have a second set of electrodes 244 and245 having an opposite electrical polarity to the first set ofelectrodes 256, 257, 258 and 259. The vibrational arm 232 has grooves235, 237, 239 and 241 constructed in a similar manner as the vibrationalarm 231. The grooves 235, 237, 239 and 241 have a third set ofelectrodes 252, 253, 254 and 255 of the same electrical polarity, andthe side surfaces of the vibrational arm 232 have a fourth set ofelectrodes 250 and 251 with the opposite electrical polarity to thethird electrodes 252, 253, 254 and 255. The electrodes disposed on thevibrational arms 231 and 232 are connected as shown in FIG. 16( b),namely, two electrode terminals of different electrical polarity K–K′are obtained.

In detail, the first set of electrodes 256, 257, 258 and 259 disposed onthe grooves 234, 236, 238 and 240 of the vibrational arm 231 have thesame electrical polarity as the fourth set of electrodes 250 and 251disposed on both side surfaces of the vibrational arm 232, while thesecond set of electrodes 244 and 245 disposed on both side surfaces ofthe vibrational arm 231 have the same electrical polarity as the thirdset of electrodes 252, 253, 254 and 255 disposed on the grooves 235,237, 239 and 241 of the arm 232. When a direct current voltage isapplied between the electrode terminals K–K′, an electric field Exoccurs along the arrow direction inside the vibrational arms 231 and232. As the electric field Ex occurs perpendicular to the electrodesdisposed on the vibrational arms, as shown in the arrow signs, theelectric field Ex has a very large value and a large distortion occursat the vibrational arms. As a result, a quartz crystal tuning forkresonator capable of vibrating in a flexural mode is obtained with asmall series resistance R₁ and a high quality factor Q because even whenminiaturized there is a very large electromechanical transformationefficiency for the resonator.

FIG. 17 shows a plan view of a quartz crystal unit of a first embodimentof the present invention and omitting a lid. The quartz crystal unit 160comprises the quartz crystal tuning fork resonator 10 shown in FIG. 2, acase 165 for housing the resonator and a lid for covering an open end ofthe case (not shown here). Also, the resonator 10 has mounting arms 36and 37, each of which is mounted on a mounting portion 166 and amounting portion 167 of the case 165. In detail, an electrode disposedon the mounting arm 36 is connected to an electrode disposed on themounting portion 166 by adhesives 168 or a metal such as solder, andsimilarly, an electrode disposed on the mounting arm 37 is connected toan electrode disposed on the mounting portion 167 by adhesives 169 or ametal.

FIG. 18 shows a plan view of a quartz crystal unit of a secondembodiment of the present invention and omitting a lid. The quartzcrystal unit 170 comprises the quartz crystal tuning fork resonator 50shown in FIG. 5, a case 175 for housing the resonator and a lid forcovering an open end of the case (not shown here). Namely, the resonatorcomprises vibrational arms and a base portion. Also, the case 175 hasmounting portions 176 and 177 and the resonator 50 has a mounting arm 77protruding from the base portion, which is mounted on the mountingportion 177 of the case 175. In detail, an electrode disposed on themounting arm 77 is connected to an electrode disposed on the mountingportion 177 by adhesives 179 or a metal such as solder, and similarly,an electrode disposed on the base portion of the resonator 50 isconnected to an electrode disposed on the mounting portion 176 byadhesives 178 or a metal such as solder.

FIG. 19 shows a cross-sectional view of a quartz crystal unit of a thirdembodiment of the present invention. The quartz crystal unit 180comprises a contour mode quartz crystal resonator 185 or a thicknessshear mode quartz crystal resonator 185, a case 181 and a lid 182. Inmore detail, the resonator 185 is mounted on a mounting portion 184 ofthe case 181 by conductive adhesives 76 or solder. Also, the case 181and the lid 182 are connected through a connecting member 183. Forexample, the contour mode resonator 185 in this embodiment is the sameresonator as one of the quartz crystal tuning fork resonators 10, 50,90, 110, 120, 130, 140, 220 and 230 described in detail in FIG. 2–FIG.11 and FIG. 15–FIG. 16.

In this embodiment, circuit elements are connected at outside of thequartz crystal unit to get a quartz crystal oscillator. Namely, only thequartz crystal tuning fork resonator is housed in the unit and also, itis housed in the unit in vacuum. In this embodiment, the quartz crystalunit of a surface mounting type is shown, but the quartz crystal tuningfork resonator may be housed in a tubular type, namely, a quartz crystalunit of the tubular type. In addition, the quartz crystal unit of thepresent invention includes any shape of a quartz crystal unit comprisinga quartz crystal resonator, a case and a lid to house the quartz crystalresonator in vacuum.

Also, instead of the quartz crystal tuning fork resonator and thethickness shear mode quartz crystal resonator, one of alength-extensional mode quartz crystal resonator, a width-extensionalmode quartz crystal resonator and a Lame mode quartz crystal resonatorwhich are a contour mode quartz crystal resonator, respectively, or aSAW (Surface Acoustic Wave) resonator or a piezoelectric resonator forsensing angular velocity (angular velocity sensor) made of quartzcrystal or ceramics may be housed in the unit.

In addition, a member of the case and the lid is ceramics or glass and ametal or glass, respectively, and a connecting member is a metal orglass with low melting point. Also, a relationship of the resonator, thecase and the lid described in this embodiment is applied to a quartzcrystal oscillator of the present invention which will be described inFIG. 20.

FIG. 20 shows a cross-sectional view of a quartz crystal oscillator of afirst embodiment of the present invention. The quartz crystal oscillator190 comprises a quartz crystal oscillating circuit, a case 191 and a lid192. In this embodiment, circuit elements constructing the oscillatingcircuit are housed in a quartz crystal unit comprising a contour modequartz crystal resonator 195 or a thickness shear mode quartz crystalresonator 195, the case 191 and the lid 192. Also, the oscillatingcircuit of this embodiment comprises an amplifier 197 including afeedback resistor, the resonator 195, a plurality of capacitors (notshown here) and a drain resistor (not shown here), and a CMOS inverteris used as the amplifier 197.

In addition, in this embodiment, the resonator 195 is mounted on amounting portion 194 of the case 191 by conductive adhesives 196 orsolder. As described above, the amplifier 197 is housed in the quartzcrystal unit and mounted on the case 191. Also, the case 191 and the lid192 are connected through a connecting member 193.

FIG. 21 shows a diagram of an embodiment of a quartz crystal oscillatingcircuit constructing a quartz crystal oscillator of the presentinvention. The quartz crystal oscillating circuit 201 comprises anamplifier (CMOS inverter) 202, a feedback resistor 204, a drain resistor207, a plurality of capacitors 205, 206 and a quartz crystal resonator203. Namely, the oscillating circuit 201 comprises an amplificationcircuit 208 having the amplifier 202 and the feedback resistor 204, anda feedback circuit 209 having the drain resistor 207, the capacitors205, 206 and the quartz crystal resonator 203. The quartz crystalresonator 203 is one of the resonators already described above. Forexample, when the resonator 203 is a quartz crystal tuning forkresonator capable of vibrating in a flexural mod, an output signal ofthe oscillating circuit 201 is outputted through a buffer circuit (notshown in FIG. 21), and is an oscillating frequency of a fundamental modeof vibration of the resonator.

In other words, the oscillation frequency of the fundamental mode ofvibration of the quartz crystal tuning fork resonator is outputted fromthe oscillating circuit through the buffer circuit as an output signal.According to the present invention, a nominal frequency of thefundamental mode of vibration of the quartz crystal tuning forkresonator is within a range of 10 kHz to 200 kHz. Especially, afrequency of 32.768 kHz is very available for use in an electronicapparatus of the present invention. In general, the output signal has anoscillation frequency which is within a range of −100 PPM to +100 PPM tothe nominal frequency, e.g. 32.768 kHz.

In more detail, the quartz crystal oscillator in this example comprisesa quartz crystal oscillating circuit and a buffer circuit, namely, thequartz crystal oscillating circuit comprises an amplification circuitand a feedback circuit, and the amplification circuit comprises anamplifier (CMOS inverter) and a feedback resistor and the feedbackcircuit comprises a quartz crystal tuning fork resonator capable ofvibrating in a flexural mode, a drain resistor and a plurality ofcapacitors. Also, the quartz crystal tuning fork resonator alreadydescribed in FIG. 2–FIG. 11 and FIG. 15–FIG. 16 is used in a quartzcrystal oscillator of the present invention. Instead of the quartzcrystal tuning fork resonator, an another contour mode quartz crystalresonator such as a length-extensional mode quartz crystal resonator, awidth-extensional mode quartz crystal resonator and a Lame mode quartzcrystal resonator or a thickness shear mode quartz crystal resonator ora resonator for sensing angular velocity may be used.

FIG. 22 shows a diagram of the feedback circuit of FIG. 21. In thisembodiment, the feedback circuit has a quartz crystal tuning forkresonator capable of vibrating in a flexural mode. Now, when angularfrequency ω_(i) of the quartz crystal tuning fork resonator 203, aresistance value R_(d) of the drain resistor 207, capacitance valuesC_(g), C_(d) of the capacitors 205, 206, crystal impedance R_(ei) of thequartz crystal resonator 203, an input voltage V₁, and an output voltageV₂ are taken, a feedback rate β_(i) is defined byβ_(i)=|V₂|_(i)/|V₁|_(i), where i shows a vibration order, for example,when i=1 and 2, β₁ and β₂ are a feedback rate for a fundamental mode ofvibration and a second overtone mode of vibration of the resonator,respectively.

In addition, load capacitance C_(L) is given byC_(L)=C_(g)C_(d)/(C_(g)+C_(d)), and when C_(g)=C_(d)=C_(gd) andR_(d)>>R_(ei), the feedback rate β_(i) is given by β_(i)=1/(1+kC_(L) ²),where k is expressed by a function of ω_(i), R_(d) and R_(ei). Also,R_(ei) is approximately equal to series resistance R_(i) of theresonator.

Thus, it is easily understood from a relationship of the feedback rateβ_(i) and load capacitance C_(L) that the feedback rate of a resonancefrequency for the fundamental mode of vibration and the overtone modevibration becomes large, respectively, according to decrease of loadcapacitance C_(L). Therefore, when C_(L) has a small value, anoscillation of the overtone mode occurs very easily, instead of that ofthe fundamental mode. This is the reason why maximum amplitude of theovertone mode of vibration becomes smaller than that of the fundamentalmode of vibration, and the oscillation of the overtone mode satisfies anamplitude condition and a phase condition simultaneously which are thecontinuous condition of an oscillation in an oscillating circuit.

As it is also one object of the present invention to provide a quartzcrystal oscillator having a flexural mode, quartz crystal tuning forkresonator, capable of vibrating in a fundamental mode and having afrequency of high stability (high time accuracy) of an output signal,and having reduced electric current consumption, load capacitance C_(L)is less than 25 pF in this embodiment to reduce electric currentconsumption. To get much reduced electric current consumption, C_(L) ispreferably less than 15 pF because the electric current consumption isproportional to C_(L).

In addition, in order to suppress a second overtone mode of vibration ofthe resonator and to obtain a quartz crystal oscillator having an outputsignal of an oscillation frequency of a fundamental mode of vibration ofthe resonator, the quartz crystal oscillator comprising an amplificationcircuit and a feedback circuit is constructed so that it satisfies arelationship of α₁/α₂>β₂/β₁ and α₁β₁>1, where α₁ and α₂ are,respectively, an amplification rate of the fundamental mode of vibrationand the second overtone mode of vibration of the amplification circuit,and β₁ and β₂ are, respectively, a feedback rate of the fundamental modeof vibration and the second overtone mode of vibration of the feedbackcircuit.

In other words, the quartz crystal oscillator is constructed so that aratio of the amplification rate α₁ of the fundamental mode of vibrationand the amplification rate α₂ of the second overtone mode of vibrationof the amplification circuit is greater than that of the feedback rateβ₂ of the second overtone mode of vibration and the feedback rate β₁ ofthe fundamental mode vibration of the feedback circuit, and also aproduct of the amplification rate α₁ and the feedback rate β₁ of thefundamental mode of vibration is greater than 1. A description of afrequency of high stability in the quartz crystal oscillator will beperformed later.

Also, characteristics of the amplifier of the amplification circuitconstructing the quartz crystal oscillating circuit of this embodimentcan be expressed by negative resistance −RL_(i). For example, when i=1,negative resistance −RL₁ is for a fundamental mode of vibration of theresonator and when i=2, negative resistance −RL₂ is for a secondovertone mode of vibration of the resonator. In this embodiment, thequartz crystal oscillating circuit is constructed so that a ratio of anabsolute value of negative resistance −RL₁ of the fundamental mode ofvibration of the amplification circuit and series resistance R₁ of thefundamental mode of vibration is greater than that of an absolute valueof negative resistance −RL₂ of the second overtone mode of vibration ofthe amplification circuit and series resistance R₂ of the secondovertone mode of vibration.

That is, the oscillating circuit is constructed so that it satisfies arelationship of |−RL₁|/R₁>|−RL₂|/R₂. By constructing the oscillatingcircuit like this, an oscillation of the second overtone mode can besuppressed, as a result of which a frequency of oscillation of thefundamental mode of vibration can be output as an output signal becausethe oscillation of the fundamental mode generates easily in theoscillating circuit. In more detail, an absolute value of the negativeresistance −RL₁ is within a range of 60 kΩ to 285 kΩ, and an absolutevalue of the negative resistance −RL₂ is less than 105 kΩ to suppressthe frequency of oscillation of the second overtone mode of the quartzcrystal tuning fork resonator in the oscillating circuit and to obtainthe frequency of oscillation of the fundamental mode thereof.

In this embodiment, a quartz crystal tuning fork resonator is used, but,instead of the tuning fork resonator, an another contour mode quartzcrystal resonator such as a width-extensional mode quartz crystalresonator, a length-extensional mode quartz crystal resonator and a Lamemode quartz crystal resonator may be used in a quartz crystal oscillatorof the present invention. In this case, a principal vibration of thecontour mode quartz crystal resonator is outputted from an oscillatingcircuit constructing the quartz crystal oscillator through a buffercircuit. In order to suppress a sub-vibration of the contour mode quartzcrystal resonator, the quartz crystal oscillator comprising anamplification circuit and a feedback circuit is constructed so that itsatisfies a relationship of α_(p)/α_(s)>β_(s)/β_(p) and α_(p)β_(p)>1,where α_(p) and α_(s) are, respectively, an amplification rate of theprincipal vibration and the sub-vibration of the amplification circuit,and β_(p) and β_(s) are, respectively, a feedback rate of the principalvibration and the sub-vibration of the feedback circuit.

Similar to the quartz crystal tuning fork resonator, for the contourmode quartz crystal resonator, a quartz crystal oscillating circuit ofthe present invention is constructed so that a ratio of an absolutevalue of negative resistance −RL_(p) of the principal vibration of theamplification circuit and a series resistance R_(p) of the principalvibration is greater than that of an absolute value of negativeresistance −RL_(s) of the sub-vibration of the amplification circuit anda series resistance R_(s) of the sub-vibration. That is, the oscillatingcircuit is constructed so that it satisfies a relationship of|−RL_(p)|/R_(p)>|−RL_(s)|/R_(s). By constructing the oscillating circuitlike this, an oscillation of the sub-vibration can be suppressed, as aresult of which a frequency of oscillation of the principal vibrationcan be outputted as an output signal because the oscillation of theprincipal vibration generates easily in the oscillating circuit. Inaddition, the principal vibration and the sub-vibration have the samemode of vibration and a different order of vibration each other.

FIG. 23 shows a block diagram of an embodiment of an electronicapparatus of the present invention, and illustrating the diagram of afacsimile apparatus. As shown in FIG. 23, the apparatus generallycomprises a modem, a phonetic circuit, a timepiece circuit, a printingportion, a taking portion, an operation portion and a display portion.In this principle, perception and scanning of reflection light of lightprojected on manuscript in the taking portion are performed byCCD(Charge Coupled Device), in addition, light and shade of thereflection light are transformed into a digital signal, and the signalis modulated by the modem and is sent to a phone line(communicationline). Also, in a receiving side, a received signal is demodulated bythe modem and is printed on a paper in the print portion bysynchronizing the received signal with a signal of a sending side.

In FIG. 23, a quartz crystal resonator is used as a CPU clock of thecontrol portion and the printing portion, as a clock of the phoneticcircuit and the modem, and as a time standard of the timepiece. Namely,the resonator constructs a quartz crystal oscillator and an outputsignal of the oscillator is used. Like this, a plurality of oscillatorsis used for the electronic apparatus. For example, it is used as asignal to display time at the display portion. In this case, a quartzcrystal tuning fork resonator, capable of vibrating in a flexural modeis generally used, and e.g. as the CPU clock, a contour mode quartzcrystal resonator such as a length-extensional mode quartz crystalresonator, a width-extensional mode quartz crystal resonator and a Lamemode quartz crystal resonator or a thickness shear mode quartz crystalresonator is used. To get the facsimile apparatus of this embodimentwhich operates normally, an accuracy signal of output of the oscillatoris required for the facsimile apparatus, which is one of the electronicapparatus of the present invention. Also, a digital display and ananalogue display are included in the display of the present invention.

In this embodiment, though the facsimile apparatus is shown as anexample of an electronic apparatus of the present invention, the presentinvention is not limited to this, namely, the present invention includesall electronic apparatus, each of which comprises a quartz crystaloscillator and a display portion at least, for example, cellar phones,telephones, a TV set, cameras, a video set, video cameras, pagers,personal computers, printers, CD players, MD players, electronic musicalinstruments, car navigators, car electronics, timepieces, IC cards andso forth. In addition, the electronic apparatus may have an anotheroscillator comprising a piezoelectric resonator for sensing angularvelocity made of quartz crystal, ceramics, lithium tantalite and lithiumniobate.

Thus, the electronic apparatus of the present invention comprising adisplay portion and a quartz crystal oscillator at least may operatenormally because the quartz crystal oscillator comprises the quartzcrystal oscillating circuit with a frequency of high stability, namely,a frequency of high reliability.

Moreover, capacitance ratios r₁ and r₂ of a flexural mode, quartzcrystal tuning fork resonator are given by r₁=C₀/C₁ and r₂=C₀/C₂,respectively, where C₀ is shunt capacitance in an electrical equivalentcircuit of the resonator, and C₁ and C₂ are, respectively, motionalcapacitance of a fundamental mode of vibration and a second overtonemode of vibration in the electrical equivalent circuit of the resonator.In addition, the resonator has a quality factor Q₁ for the fundamentalmode of vibration and a quality factor Q₂ for the second overtone modeof vibration.

In detail, the tuning fork resonator of this embodiment is provided sothat the influence on resonance frequency of the fundamental mode ofvibration by the shunt capacitance becomes smaller than that of thesecond overtone mode of vibration by the shunt capacitance, namely, sothat it satisfies a relationship of S₁=r₁/2Q₁ ²<S₂=r₂/2Q₂ ², preferably,S₁<S₂/2, where S₁ and S₂ are called “a stable factor of frequency” ofthe fundamental mode of vibration and the second overtone mode ofvibration. As a result, the tuning fork resonator, capable of vibratingin the fundamental mode and having a frequency of high stability can beprovided because the influence on the resonance frequency of thefundamental mode of vibration by the shunt capacitance C₀ is asextremely small as it can be ignored. In this embodiment S₂ has a valuegreater than 0.13×10⁻⁶ to suppress the second overtone mode of vibrationof the resonator.

In addition, as described above, it will be easily understood that thequartz crystal resonator comprising vibrational arms and a base portion,according to the present invention, may have outstanding effects.Similar to this, the quartz crystal unit and the quartz crystaloscillator, according to the present invention, may have alsooutstanding effects. In addition, the electronic apparatus comprisingthe quartz crystal oscillator comprising the quartz crystal oscillatingcircuit having the quartz crystal tuning fork resonator, capable ofvibrating in a flexural mode, and having novel shapes, novel electrodeconstruction and excellent electrical characteristics, according to thepresent invention, may have the outstanding effects. Similar to this, itwill be easily understood that the electronic apparatus comprising thequartz crystal oscillator comprising the quartz crystal oscillatingcircuit having the another contour mode quartz crystal resonator or thethickness shear mode quartz crystal resonator or the resonator forsensing angular velocity, according to the present invention, may havealso the outstanding effect. In addition to this, while the presentinvention has been shown and described with reference to preferredembodiments thereof, it will be understood by those skilled in the artthat the changes in shape and electrode construction can be made thereinwithout departing from the spirit and scope of the present invention.

1. A quartz crystal resonator comprising: a base portion and a pluralityof vibrational arms connected to the base portion; wherein the quartzcrystal resonator has a piezoelectric constant e′₁₂ in the range of 0.1C/m² to 0.19 C/m² in the absolute value.
 2. A quartz crystal resonatoraccording to claim 1; wherein the resonator according to claim 1;wherein the quartz crystal resonator has a cutting angle in the range ofZYlwt(−20° to +20°)/(−25° to +25°)/(−18° to +18°).
 3. A quartz crystalresonator according to claim 1; wherein the vibrational arms have afirst vibrational arm and a second vibrational arm; each of the firstand second vibrational arms having a first main surface and a secondmain surface opposite the first main surface; wherein at least onegroove is formed in each of the first and second main surfaces of eachof the first second vibrational arms; wherein a length of at least oneof the grooves formed in the first and second main surfaces of each ofthe first and second vibrational arms is within a range of 30% to 80% ofa length of each of the first and second vibrational arms and is withina range of 0.45mm to 1.25mm; and wherein an electrode is disposed in theat least one groove formed in each of the first and second main surfacesof each of the first and second vibrational arms so that a capacitanceratio r1 of a fundamental mode of vibration of the quartz crystalresonator is less than a capacitance ratio r₂ of a second overtone modeof vibration thereof.
 4. A quartz crystal resonator according to claim1; wherein the quartz crystal resonator comprises a quartz crystaltuning fork resonator having a tuning fork base and first and secondtuning fork arms connected to the tuning fork base, each of the firstand second tuning fork arms having a first main surface and a secondmain surface opposite the first main surface; wherein at least onegroove is formed in each of the first and second main surfaces of eachof the first and second tuning fork arms; wherein a length of the tuningfork base is within a range of 0.015mm to 0.49mm and an overall lengthof the quartz crystal tuning fork resonator having the tuning fork baseand the first and second tuning fork arms connected to the tuning forkbase is within a range of 1.02mm to 1.95mm; and wherein an electrode isdisposed in the at least one groove formed in each of the first andsecond main surfaces of each of the first and second vibrational arms sothat a capacitance ratio r1 of a fundamental mode of vibration of thequartz crystal resonator is less than a capacitance ratio r2 of a secondovertone mode of vibration thereof.
 5. A quartz crystal resonatoraccording to claim 1; wherein the quartz crystal resonator comprises aquartz crystal tuning fork resonator having a tuning fork base and aplurality of tuning fork arms connected to the tuning fork base; whereinthe tuning fork arms have a first tuning fork arm and a second tuningfork arm, each of the first and second tuning fork arms having a firstmain surface and a second main surface opposite the first main surface;wherein at least one groove is formed in each of the first and secondmain surfaces of each of the first and second tuning fork arms; andwherein at least one mounting arm is connected to the tuning fork basethrough a connecting portion so that a length of the at least onemounting arm is within a range of 0.3 mm to 1.85 mm.
 6. A quartz crystalunit comprising: a quartz crystal resonator, capable of vibrating in aflexural mode of an inverse phase and having a base portion and aplurality of vibrational arms connected to the base portion; a case forhousing the quartz crystal resonator; and a lid for covering an open endof the case; wherein the quartz crystal resonator has a cutting angle inthe range of ZYlwt (−20°. to 20°.)/(−25°. to 25°.)/(−18°. to −18°.) anda piezoelectric constant e'₁₂ of the quartz crystal resonator is withina range of 0.1 C/m² to 0.19 C/m² in the absolute value.
 7. A quartzcrystal unit according to claim 6; wherein the vibrational arms have afirst vibrational arm and a second vibrational arm, each of the firstand second vibrational arms having a first main surface and a secondmain surface opposite the first main surface; and wherein at least onegroove is formed in at least one of the first and second main surfacesof each of the first and second vibrational arms.
 8. A quartz crystalunit according to claim 7; wherein the at least one groove formed in atleast one of the first and second main surfaces of each of the first andsecond vibrational arms comprises a plurality of grooves having a firstgroove and a second groove in at least one of the first and second mainsurfaces of each of the first and second vibrational arms; wherein thefirst and second grooves are formed in the width direction of thecorresponding one of the first and second vibrational arms; wherein awidth of the first groove formed in at least one of the first and secondmain surfaces of each of the first and second vibrational arms isgreater than or equal to a distance in the width direction of the firstgroove measured from an outer edge of the first groove to an outer edgeof the corresponding one of the first and second vibrational arms; andwherein a width of the second groove formed in at least one of the firstand second main surfaces of each of the first and second vibrationalarms is greater than or equal to a distance in the width direction ofthe second groove measured from an outer edge of the second groove to anouter edge of the corresponding one of the first and second vibrationalarms.
 9. A quartz crystal unit according to claim 7; wherein the atleast one groove formed in at least one of the first and second mainsurfaces of each of the first and second vibrational arms comprises aplurality of grooves having a first groove and a second groove in atleast one of the first and second main surfaces of each of the first andsecond vibrational arms; wherein the first and second grooves are formedin the width direction of the corresponding one of the first and secondvibrational arms; and wherein a width of at least one of the first andsecond grooves is greater than or equal to a distance in the widthdirection of the first and second grooves measured from an outer edge ofthe first groove to an outer edge of the second groove.
 10. A quartzcrystal unit according to claim 9; wherein the distance in the widthdirection of the first and second grooves measured from the outer edgeof the first groove to the outer edge of the second groove is less than0.05 mm.
 11. A quartz crystal unit according to claim 7; wherein the atleast one groove formed in at least one of the first and second mainsurfaces of each of the first and second vibrational arms comprises aplurality of grooves having a first groove and a second groove in atleast one of the first and second main surfaces of each of the first andsecond vibrational arms; and wherein the first and second grooves areformed in the width direction of the corresponding one of the first andsecond vibrational arms so that a width of at least one of the first andsecond grooves is within a range of 0.015 mm to 0.05 mm.
 12. A quartzcrystal unit according to claim 7; wherein the at least one grooveformed in at least one of the first and second main surfaces of each ofthe first and second vibrational arms comprises a plurality of grooveshaving a first groove and a second groove in at least one of the firstand second main surfaces of each of the first and second vibrationalarms; wherein the first and second grooves are formed in the widthdirection of the corresponding one of the first and second vibrationalarms; wherein each of the first and second vibrational arms has acentral linear portion, a first outer edge and a second outer edgeopposite the first outer edge; wherein the first groove is formedbetween the central linear portion and the first outer edge and thesecond groove is formed between the central linear portion and thesecond outer edge; and wherein an electrode is disposed in each of thefirst and second grooves in each of the first and second main surfacesof each of the first and second vibrational arms so that a seriesresistance R₁ of a fundamental mode of vibration of the quartz crystalresonator is less than a series resistance of a second overtone mode ofvibration thereof and a capacitance ratio r₁ of the fundamental mode ofvibration of the quartz crystal resonator is less than a capacitanceratio r₂ of the second overtone mode of vibration thereof.
 13. A quartzcrystal unit according to claim 7; wherein a mounting arm protrudes fromthe base portion; and wherein the mounting arm is mounted on a mountingportion of the case for housing the quartz crystal resonator.
 14. Aquartz crystal unit according to claim 13; wherein the quartz crystalresonator comprises a quartz crystal tuning fork resonator, capable ofvibrating in a flexural mode and having a quartz crystal tuning forkbase, first and second quartz crystal tuning fork arms connected to thequartz crystal tuning fork base and at least one groove formed in eachof opposite main surfaces of each of the first and second quartz crystaltuning fork arms; wherein a mounting arm protrudes from the quartzcrystal tuning fork base and is formed between the first and secondquartz crystal tuning fork arms; and wherein the mounting arm is mountedon a mounting portion of a case for housing the quartz crystal tuningfork resonator.
 15. A quartz crystal unit according to claim 13; whereinthe quartz crystal resonator comprises a quartz crystal tuning forkresonator, capable of vibrating in a flexural mode and having a quartzcrystal tuning fork base, first and second quartz crystal tuning forkarms connected to the quartz crystal tuning fork base and at least onegroove formed in each of opposite main surfaces of each of the first andsecond quartz crystal tuning fork arms; wherein a mounting arm protrudesfrom the quartz crystal tuning fork base so that the mounting armextends in a common direction with the first and second quartz crystaltuning fork arms outside the first and second quartz crystal tuning forkarms; and wherein the mounting arm is mounted on a mounting portion of acase for housing the quartz crystal tuning fork resonator.
 16. A quartzcrystal unit according to claim 6; wherein the quartz crystal resonatorcomprises a quartz crystal tuning fork resonator having a tuning forkbase and first and second tuning fork arms connected to the tuning forkbase, each of the first and second tuning fork arms having a first mainsurface and a second main surface opposite the first main surface;wherein at least one groove is formed in each of the first and secondmain surfaces of each of the first and second tuning fork arms; whereinat least one mounting arm is connected to the tuning fork base through aconnecting portion and is mounted on a mounting portion of a case forhousing the quartz crystal tuning fork resonator; wherein a length ofthe tuning fork base is less than 0.5 mm and an overall length of thequartz crystal tuning fork resonator having the tuning fork base and thefirst and second tuning fork arms connected to the tuning fork base iswithin a range of 1.02 mm to 1.95 mm; and wherein an electrode isdisposed in the at least one groove formed in each of the first andsecond main surfaces of each of the first and second tuning fork arms sothat a capacitance ratio r₁ of a fundamental mode of vibration of thequartz crystal tuning fork resonator is less than a capacitance ratio r2of a second overtone mode of vibration thereof.
 17. A quartz crystalunit according to claim 6; wherein the quartz crystal resonatorcomprises a quartz crystal tuning fork resonator, capable of vibratingin a flexural mode and having a quartz crystal tuning fork base, firstand second quartz crystal tuning fork arms connected to the quartzcrystal tuning fork base and a groove formed in each of opposite mainsurfaces of each of the first and second quartz crystal tuning forkarms; wherein each of the first and second quartz crystal tuning forkarms has a plurality of different widths having a first width and asecond width greater than the first width; and wherein the groove formedin each of the opposite main surfaces of each of the first and secondquartz crystal tuning fork arms is formed in at least one of the firstand second width directions of the corresponding one of the first andsecond quartz crystal tuning fork arms.
 18. A quartz crystal unitaccording to claim 17; wherein at least one mounting arm is connected tothe quartz crystal tuning fork base through a connecting portion; andwherein the at least one mounting arm is mounted on a mounting portionof a case for housing the quartz crystal tuning fork resonator.
 19. Aquartz crystal unit according to claim 17; wherein the groove formed ineach of the opposite main surfaces of each of the first and secondquartz crystal tuning fork arms is formed in the first width directionof the corresponding one of the first and second quartz crystal tuningfork arms.
 20. A quartz crystal unit according to claim 6; wherein thequartz crystal resonator comprises a quartz crystal tuning forkresonator having a tuning fork base and first and second tuning forkarms connected to the tuning fork base, each of the first and secondtuning fork arms having a first main surface and a second main surfaceopposite the first main surface; wherein a plurality of grooves having afirst groove and a second groove are formed in each of the first andsecond main surfaces of each of the first and second tuning fork arms;wherein the first and second grooves are formed in the width directionof the corresponding one of the first and second tuning fork arms;wherein an electrode is disposed in each of the first and second groovesin each of the first and second main surfaces of each of the first andsecond tuning fork arms so that a series resistance of a fundamentalmode of vibration of the quartz crystal tuning fork resonator is lessthan a series resistance R₁ of a second overtone mode of vibrationthereof; and further comprising an amplification circuit having a CMOSinverter and a feedback resistor, and a feedback circuit having thequartz crystal tuning fork resonator capable of vibrating in a flexuralmode, a plurality of capacitors and a drain resistor; wherein the quartzcrystal tuning fork resonator is electrically connected to the CMOSinverter and the feedback resistor of the amplification circuit and tothe capacitors and the drain resistor of the feedback circuit to get aquartz crystal oscillating circuit.
 21. A quartz crystal unit accordingto claim 7; wherein the at least one groove formed in at least one ofthe first and second main surfaces of each of the first and secondvibrational arms comprises a groove in each of the first and second mainsurfaces of each of the first and second vibrational arms; wherein eachof the first and second main surfaces of each of the first and secondvibrational arms has a central linear portion; and wherein at least oneof the grooves in the first and second main surfaces of each of thefirst and second vibrational arms is formed in the central linearportion so that a distance in the width direction of the at least onegroove measured from an outer edge of the at least one groove to anouter edge of the corresponding one of the first and second vibrationalarms is less than 0.015 mm.
 22. An electronic apparatus comprising: adisplay portion; and a plurality of oscillators, one of the oscillatorsbeing a quartz crystal oscillator comprising a quartz crystaloscillating circuit comprising an amplification circuit having a CMOSinverter and a feedback resistor, and a feedback circuit having a quartzcrystal resonator capable of vibrating in a flexural mode, a pluralityof capacitors and a drain resistor, the quartz crystal resonator havinga base portion and a plurality of vibrational arms connected to the baseportion, the quartz crystal resonator being housed in a packagecomprising a case for housing the quartz crystal resonator and a lid forcovering an open end of the case; and wherein the quartz crystalresonator has a cutting angle in the range of ZYlwt (−20to −20°.)/(−25to−25°.)/(−18°. to −18°.) and a piezoelectric constant e12 of the quartzcrystal resonator is within a range of 0.1 C/m² to 0.19 C/m² in theabsolute value.
 23. An electronic apparatus according to claim 22;wherein one of the oscillators is a quartz crystal oscillator comprisingan oscillating circuit comprising an amplification circuit having a CMOSinverter and a feedback resistor, and a feedback circuit having athickness shear mode quartz crystal resonator, a plurality of capacitorsand a drain resistor, a pair of electrodes being disposed on avibrational portion of the thickness shear mode quartz crystalresonator, the thickness shear mode quartz crystal resonator having acutting angle in the range of ZYlwt(−50°. to −5°.)/±(37°. to 58°.)/(85°.to 95°.) and a piezoelectric constant e'₃₄ in the range of 0.055 C/m² to0.14 C/m² in the absolute value.
 24. An electronic apparatus accordingto claim 22; wherein one of the oscillators is a quartz crystaloscillator comprising an oscillating circuit comprising an amplificationcircuit having a CMOS inverter and a feedback resistor, and a feedbackcircuit having a Lame mode quartz crystal resonator, a plurality ofcapacitors and a drain resistor, a pair of electrodes being disposed ona vibrational portion of the Lame mode quartz crystal resonator, theLame mode quartz crystal resonator having a cutting angle in the rangeof ZYlwt(−5°. to +5°.)/±(35°. to 60°.)/±(40°. to 500°.) and apiezoelectric constant e'₃₂ in the range of 0.05 C/m² to 0.13 C/m² inthe absolute value.
 25. An electronic apparatus according to claim 22;wherein the plurality of vibrational arms have a first vibrational armand a second vibrational arm, each of the first and second vibrationalarms having a first main surface and a second main surface opposite thefirst main surface; wherein at least one groove is formed in at leastone of the first and second main surfaces of each of the first andsecond vibrational arms; and wherein an output signal of the quartzcrystal oscillating circuit comprising the quartz crystal resonatorhaving the first and second clock signal for use in operation of theelectronic apparatus to display time information at the display portion.26. An electronic apparatus according to claim 25; wherein a mountingarm protrudes from the base portion; and wherein the mounting arm ismounted on a mounting portion of the case for housing the quartz crystalresonator.
 27. An electronic apparatus according to claim 26; wherein alength of the mounting arm is within a range of 0.3 mm to 1.85 mm. 28.An electronic apparatus according to claim 25; wherein the at least onegroove formed in at least one of the first and second main surfaces ofeach of the first and second vibrational arms comprises a plurality ofgrooves having a first groove and a second groove in at least one of thefirst and second main surfaces of each of the first and secondvibrational arms; wherein the first and second grooves are formed in thewidth direction of the corresponding one of the first and secondvibrational arms; wherein a width of the first groove formed in at leastone of the first and second main surfaces of each of the first andsecond vibrational arms is greater than or equal to a distance in thewidth direction of the first groove measured from an outer edge of thefirst groove to an outer edge of the corresponding one of the first andsecond vibrational arms; and wherein a width of the second groove formedin at least one of the first and second main surfaces of each of thefirst and second vibrational arms is greater than or equal to a distancein the width direction of the second groove measured from an outer edgeof the second groove to an outer edge of the corresponding one of thefirst and second vibrational arms.
 29. An electronic apparatus accordingto claim 25; wherein the at least one groove formed in at least one ofthe first and second main surfaces of each of the first and secondvibrational arms comprises a plurality of grooves having a first grooveand a second groove in at least one of the first and second mainsurfaces of each of the first and second vibrational arms; wherein thefirst and second grooves are formed in the width direction of thecorresponding one of the first and second vibrational arms; and whereina width of at least one of the first and second grooves is greater thanor equal to a distance in the width direction of the first and secondgrooves measured from an outer edge of the first groove to an outer edgeof the second groove.
 30. An electronic apparatus according to claim 25;wherein the quartz crystal resonator has a fundamental mode of vibrationand a second overtone mode of vibration capable of vibrating in aflexural mode; wherein the amplification circuit of the quartz crystaloscillating circuit has negative resistances −RL₁ and −RL₂ for thefundamental mode of vibration and the second overtone mode of vibrationof the quartz crystal resonator, an absolute value of the negativeresistances being defined by |−RL₁|and |−RL₂|; and wherein a ratio of|−RL₁|and R₁ is greater than that of |−RL₂|and R₂, where R₁ and R₂represent a series resistance of the fundamental mode of vibration andthe second overtone mode of vibration of the quartz crystal resonator,respectively.
 31. An electronic apparatus according to claim 30; whereinthe absolute value of the negative resistance −RL₂ is less than 105 kΩ.32. An electronic apparatus according to claim 22; wherein one of theoscillators is a quartz crystal oscillator comprising a quartz crystaloscillating circuit having an amplifier, a quartz crystal unit having athickness shear mode quartz crystal resonator, a plurality of capacitorsand a plurality of resistors, a pair of electrodes being disposed on avibrational portion of the thickness shear mode quartz crystalresonator; and wherein an output signal of the quartz crystaloscillating circuit having the thickness shear mode quartz crystalresonator is a clock signal for use in operation of the electronicapparatus.
 33. An electronic apparatus according to claim 32; whereinthe quartz crystal resonator comprises a quartz crystal tuning forkresonator having a tuning fork base and first and section d tuning forkarms connected to the tuning fork base, each of the first and secondtuning fork arms having a first main surface and a second main surfaceopposite the first main surface and opposite side surfaces having afirst side surface and a second side surface; wherein a groove having afirst stepped portion and a second stepped portion is formed in each ofthe first and second main surfaces of each of the first and secondtuning fork arms so that the groove formed in each of the first andsecond main surfaces of each of the first and second tuning fork arms isformed in a central linear portion of each of the first and second mainsurfaces of each of the first and second tuning fork arms and a width ofthe groove formed in each of the first and second main surfaces of eachof the first and second tuning fork arms is greater than or equal to adistance in the width direction of the groove measured from an outeredge of the groove to an outer edge of the corresponding one of thefirst and second tuning fork arms; wherein a first electrode is disposedon the first stepped portion of the groove formed in each of the firstand second main surfaces of each of the first and second tuning forkarms, a second electrode being disposed on the second stepped portion ofthe groove formed in each of the first and second main surfaces of eachof the first and second tuning fork arms, and a third electrode beingdisposed on each of the first and second side surfaces of each of thefirst and second tuning fork arms, the first and second electrodes beingformed between the third electrodes; wherein the piezoelectric constante'12 is located between the third electrodes disposed on the first andsecond side surfaces of each of the first and second tuning fork arms;wherein when the piezoelectric constant e'12 located between the firstelectrode and the third electrode is defined by a piezoelectric constantand the piezoelectric constant e'₁₂ located between the second electrodeand the third electrode is defined by a piezoelectric constant e'_(12o),each of the piezoelectric constants e'_(12i) and e'_(12o) is in therange of 0.12 C/m ² to 0.19C/m² in the absolute value; and wherein aproduct of e'_(12i) and e'_(12o) is greater than
 0. 34. An electronicapparatus according to claim 32; wherein the quartz crystal resonatorcomprises a quartz crystal tuning fork resonator having a tuning forkbase and first and second tuning fork arms connected to the tuning forkbase, each of the first and second tuning fork arms having a first mainsurface and a second main surface opposite the first main surface;wherein at least one groove is formed in each of the first and secondmain surfaces of each of the first and second tuning fork arms; whereina length of the tuning fork base is less than 0.5 mm and an overalllength of the quartz crystal tuning fork resonator having the tuningfork base and the first and second tuning fork arms connected to thetuning fork base is within a range of 1.02 mm to 1.95 mm; and wherein anelectrode is disposed in the at least one groove formed in each of thefirst and second main surfaces of each of the first and second tuningfork arms so that a capacitance ratio r1 of a fundamental mode ofvibration of the quartz crystal tuning fork resonator is less than acapacitance ratio r2 of a second overtone mode of vibration thereof. 35.An electronic apparatus according to claim 32; wherein the quartzcrystal resonator comprises a quartz crystal tuning fork resonatorhaving a tuning fork base and first and second tuning fork armsconnected to the tuning fork base, each of the first and second tuningfork arms having a first main surface and a second main surface oppositethe first main surface; wherein at least one groove is formed in each ofthe first and second main surfaces of each of the first and secondtuning fork arms; wherein an overall length of the quartz crystal tuningfork resonator having the tuning fork base and the first and secondtuning fork arms connected to the tuning fork base is within a range of1.02 mm to 1.95 mm; and wherein an electrode is disposed in the at leastone groove formed in each of the first and second main surfaces of eachof the first and second tuning fork arms so that a stable factor S₁ isless than a stable factor S₂, the stable factors S₁ and S₂ being definedby r₁/2Q₁ ² and r₂/2Q₂ ², respectively, where Q1 and Q2 represent aquality factor of a fundamental mode of vibration and a second overtonemode of vibration, respectively, of the quartz crystal tuning forkresonator and r₁ and r₂ represent a capacitance ratio of the fundamentalmode of vibration and the second overtone mode of vibration,respectively, of the quartz crystal tuning fork resonator.
 36. Anelectronic apparatus according to claim 32; wherein the quartz crystalresonator comprises a quartz crystal tuning fork resonator, capable ofvibrating in a flexural mode and having a quartz crystal tuning forkbase, first and second quartz crystal tuning fork arms connected to thequartz crystal tuning fork base and a groove formed in each of oppositemain surfaces of each of the first and second quartz crystal tuning forkarms; wherein each of the first and second quartz crystal tuning forkarms has a plurality of different widths having a first width and asecond width greater than the first width; and wherein the groove formedin each of the opposite main surfaces of each of the first and secondquartz crystal tuning fork arms is formed in at least one of the firstand second width directions of the corresponding one of the first andsecond quartz crystal tuning fork arms.
 37. An electronic apparatusaccording to claim 36; wherein at least one mounting arm is connected tothe quartz crystal tuning fork base through a connecting portion; andwherein the at least one mounting arm is mounted on a mounting portionof a case for housing the quartz crystal tuning fork resonator.
 38. Anelectronic apparatus according to claim 36; wherein the groove formed ineach of the opposite main surfaces of each of the first and secondquartz crystal tuning fork arms is formed in the first width directionof the corresponding one of the first and second quartz crystal tuningfork arms.
 39. An electronic apparatus according to claim 32; whereinthe quartz crystal resonator comprises a quartz crystal tuning forkresonator having a tuning fork base and first and second tuning forkarms connected to the tuning fork base, each of the first and secondtuning fork arms having a first side surface and a second side surfaceopposite the first side surface; wherein a length of the tuning forkbase is within a range of 0.015 mm to 0.49 mm; and wherein an electrodeis disposed on each of the first and second side surfaces of each of thefirst and second tuning fork arms so that the electrode disposed on eachof the first and second side surfaces of the first tuning fork arm hasan electrical polarity opposite to an electrical polarity of theelectrode disposed on each of the first and second side surfaces of thesecond tuning fork arm and a capacitance ratio r₁ of a fundamental modeof vibration of the quartz crystal tuning fork resonator is less than acapacitance ratio r₂ of a second overtone mode of vibration thereof.